# Properties

 Label 2.111.6t5.a.b Dimension 2 Group $S_3\times C_3$ Conductor $3 \cdot 37$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $111= 3 \cdot 37$ Artin number field: Splitting field of 6.0.36963.1 defined by $f= x^{6} - 3 x^{5} + 4 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $S_3\times C_3$ Parity: Odd Determinant: 1.111.6t1.b.a Projective image: $S_3$ Projective field: Galois closure of 3.1.4107.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{2} + 7 x + 2$
Roots:
 $r_{ 1 }$ $=$ $4 a + 2 + \left(2 a + 9\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(9 a + 2\right)\cdot 11^{3} + 3\cdot 11^{4} + \left(a + 10\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 2 }$ $=$ $7 a + 6 + \left(5 a + 8\right)\cdot 11 + \left(4 a + 4\right)\cdot 11^{2} + \left(a + 2\right)\cdot 11^{3} + \left(8 a + 1\right)\cdot 11^{4} + \left(a + 3\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 3 }$ $=$ $a + 8 + 5 a\cdot 11 + \left(4 a + 7\right)\cdot 11^{2} + 4 a\cdot 11^{3} + \left(4 a + 4\right)\cdot 11^{4} + \left(3 a + 8\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 4 }$ $=$ $4 a + 1 + \left(5 a + 2\right)\cdot 11 + \left(6 a + 6\right)\cdot 11^{2} + \left(9 a + 3\right)\cdot 11^{3} + \left(2 a + 10\right)\cdot 11^{4} + \left(9 a + 1\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 5 }$ $=$ $7 a + 7 + \left(8 a + 3\right)\cdot 11 + \left(a + 10\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} + \left(9 a + 2\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 6 }$ $=$ $10 a + 1 + \left(5 a + 9\right)\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + \left(6 a + 2\right)\cdot 11^{3} + \left(6 a + 6\right)\cdot 11^{4} + \left(7 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2,6)$ $(3,5,4)$ $(1,4,6,5,2,3)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,5)(2,4)(3,6)$ $0$ $1$ $3$ $(1,6,2)(3,4,5)$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,2,6)(3,5,4)$ $2 \zeta_{3}$ $2$ $3$ $(1,2,6)$ $\zeta_{3} + 1$ $2$ $3$ $(1,6,2)$ $-\zeta_{3}$ $2$ $3$ $(1,6,2)(3,5,4)$ $-1$ $3$ $6$ $(1,4,6,5,2,3)$ $0$ $3$ $6$ $(1,3,2,5,6,4)$ $0$
The blue line marks the conjugacy class containing complex conjugation.